Can someone help me identify this .22 rifle?

Okay, I just bought my first rifle off a friend of mine, and it's a single shot, bolt action .22LR

Engraving on the top says "Eatonia 22 Long Rifle"
I know Eatonia rifles were made by cooey and bought out by winchester, but every cooey I've seen has had a serial number on it, but this one doesn't for some reason.
Also, there seems to be a chip taken out of the inner chamber where you load your .22 bullet.

All in all I have a few questions I need answered

A) Why doesn't this one have a serial number?
B) What eatonia model is it? (I can't find any identical ones online)
C) Is it safe to fire, given the chip missing from the chamber?

Serial numbers were not required on many long guns in the US until the 1968 Gun Control Act, I have about a dozen Mossberg 22s that have never worn a serial number- and one that has- made for the US Military.

Ref: Chip- I try not to diagnose firearms at remote places- my crystal ball is in the shop for calibration and an oil change. HOWEVER-

Many rimfire weapons develop a spot on the chamber mouth from the firing pin hitting the mouth. Especially if dry fired. This MAY lead to problems with extraction- there is a little burr pushed out that the cartridge drags against. In severe cases, failure to support the rim may cause a fail-to-fire.

I would have a competent smith check your rifle. If there is a burr in the chamber, DO NOT FILE IT OUT. There is a technique using an "iron" to push the metal back in place (Me cheap chollie- use a Stanley nailset )

i have the exact gun, fire it all the time, my father got it when he was 12, that makes it 1931 model, it had been fired so much, i had to get the bolt replaced cause it was not a sure thing when you pulled the trigger.you have to pull the bolt back after loading, the second generation rifle, this was done automatically when you cocked the rifle. i have pellet guns with longer barrels, lol, this little gun is dead on , a long shot with the peep site, just as deadly. no serial number on it either, even took the barrel off the stock , nothing.clark